Note that Theorem REF  actually reproves the product theorem of [1]} when the base field is finitely generated and of characteristic zero. Indeed, if \(X\)  and \(Y\)  satisfy the Hilbert property (hence weak-Hilbert property) over a finitely generated field \(k\)  of characteristic zero, then \(X\times Y\)  has the weak-Hilbert property over \(k\)  by Theorem REF . However, since \(X\)  and \(Y\)  are geometrically simply-connected (by Corvaja-Zannier's theorem [2]}), it follows that \(X\times Y\)  is geometrically simply-connected (by the product property for étale fundamental groups). Thus, the smooth proper variety \(X\times Y\)  is geometrically simply-connected and has the weak-Hilbert property over \(k\) . Therefore, by definition, it has the Hilbert property over \(k\) .
